Alright, let's try to figure out this sequence problem. The sequence starts with 0, and each subsequent term is generated by alternately adding and multiplying by successive integers starting from 1. The question is asking for the first term in the sequence that's greater than 125. Hmm, okay.

First, let's make sure I understand the sequence correctly. The first term is 0. Then, the second term is obtained by adding 1 to the first term. So 0 + 1 = 1. The third term is obtained by multiplying the second term by 1. So 1 * 1 = 1. The fourth term is adding 2 to the third term: 1 + 2 = 3. The fifth term is multiplying the fourth term by 2: 3 * 2 = 6. The sixth term is adding 3 to the fifth term: 6 + 3 = 9. The seventh term is multiplying the sixth term by 3: 9 * 3 = 27. Then the eighth term would be adding 4 to 27, which is 31. The ninth term would be multiplying 31 by 4, which is 124. Wait, 124 is less than 125. Then the tenth term would be adding 5 to 124: 124 + 5 = 129. So 129 is the first term greater than 125. Is that right?

Wait, let me check again step by step to make sure I didn't skip any steps or make a mistake in the operations. Let's list out the terms with their operations:

1st term: 0 (given)

2nd term: 0 + 1 = 1 (add 1)

3rd term: 1 * 1 = 1 (multiply by 1)

4th term: 1 + 2 = 3 (add 2)

5th term: 3 * 2 = 6 (multiply by 2)

6th term: 6 + 3 = 9 (add 3)

7th term: 9 * 3 = 27 (multiply by 3)

8th term: 27 + 4 = 31 (add 4)

9th term: 31 * 4 = 124 (multiply by 4)

10th term: 124 + 5 = 129 (add 5)

11th term: 129 * 5 = 645 (multiply by 5)

Wait, so the 10th term is 129, which is the first term over 125. So the answer would be 129. But let me verify each step again because sometimes when you do this manually, it's easy to make a mistake.

Starting from term 1 to term 10:

1: 0

2: 0 +1 =1

3:1 *1=1

4:1 +2=3

5:3 *2=6

6:6 +3=9

7:9 *3=27

8:27 +4=31

9:31 *4=124

10:124 +5=129

Yes, that seems correct. So term 10 is 129, which is the first term over 125. Therefore, the answer is 129. But wait, let me check if there's any term between 9 and 10 that might be over 125. Term 9 is 124, which is just below 125. Then term 10 is 129, which is the first over. So yes, 129 is the answer.

But let me make sure the operations are alternating correctly. The problem says "alternately adding and multiplying by each successive integer beginning with 1." So starting with the second term, the operations alternate between addition and multiplication. Let's see:

Term 2: add 1

Term 3: multiply by 1

Term 4: add 2

Term 5: multiply by 2

Term 6: add 3

Term 7: multiply by 3

Term 8: add 4

Term 9: multiply by 4

Term 10: add 5

Yes, that's correct. So each time, the operation alternates, starting with addition for the second term. So the operations are: add 1, multiply by 1, add 2, multiply by 2, etc.

Another way to think about it is that for each pair of terms (even and odd), we add an odd term and then multiply by the next integer. Wait, maybe not. Let's see:

Wait, starting from term 1:

Term 1: 0

Term 2: add 1 (operation: +1)

Term 3: multiply by 1 (operation: *1)

Term 4: add 2 (operation: +2)

Term 5: multiply by 2 (operation: *2)

Term 6: add 3 (operation: +3)

Term 7: multiply by 3 (operation: *3)

Term 8: add 4 (operation: +4)

Term 9: multiply by 4 (operation: *4)

Term 10: add 5 (operation: +5)

Yes, so the operations alternate between addition and multiplication, starting with addition for the second term. Each time, the integer used in the operation increases by 1. So first addition with 1, then multiplication with 1, then addition with 2, multiplication with 2, etc.

So the operations are in the order: add 1, multiply by 1, add 2, multiply by 2, add 3, multiply by 3, etc.

Therefore, each pair of operations (add and multiply) uses the same integer, but the integer increases by 1 each time. So the sequence of operations is:

+1, *1, +2, *2, +3, *3, +4, *4, +5, *5, etc.

So that's the pattern. Therefore, the terms are calculated as:

Term 1: 0

Term 2: 0 +1 =1

Term 3: 1 *1 =1

Term 4: 1 +2 =3

Term 5: 3 *2 =6

Term 6: 6 +3 =9

Term 7: 9 *3 =27

Term 8: 27 +4 =31

Term 9: 31 *4 =124

Term 10: 124 +5 =129

Term 11: 129 *5 =645

So term 10 is 129, which is the first term over 125. Term 9 is 124, which is just below 125. Therefore, the answer is 129.

But wait, just to be absolutely thorough, let me check term 9 again. Term 8 is 31. Then term 9 is 31 multiplied by 4. 31*4 is 124. Correct. Then term 10 is 124 +5 =129. Yes. So 129 is indeed the first term over 125.

Is there any chance that between term 9 and term 10, there's a term that's over 125? But the terms are generated in order, so term 9 is 124, then term 10 is 129. So 129 is the first one over 125. Therefore, the answer is 129.

Wait, but the problem says "the first term that is greater than 125". 129 is greater than 125, so that's correct. But let me check if 129 is indeed the first term. Let's see:

Term 9: 124

Term 10: 129

So yes, term 10 is the first one over 125. Therefore, the answer is 129.

But let me just think if there's a way this could be wrong. For example, maybe the operations are misinterpreted. For instance, does the first term after adding 1, then multiply by 1, then add 2, etc., which would be the case. So each operation is applied to the previous term. So each operation is performed on the result of the previous operation. So the sequence is built step by step, each term depending on the previous one.

Therefore, the steps are correct. So the answer should be 129.

But perhaps the problem is in the way the operations are applied. Let me check again:

Term 1: 0

Term 2: Term1 +1 =0 +1=1

Term3: Term2 *1=1*1=1

Term4: Term3 +2=1 +2=3

Term5: Term4 *2=3*2=6

Term6: Term5 +3=6 +3=9

Term7: Term6 *3=9*3=27

Term8: Term7 +4=27 +4=31

Term9: Term8 *4=31*4=124

Term10: Term9 +5=124 +5=129

Yes, that's correct.

Alternatively, perhaps the problem is that the operations are applied to the term before, not the term after. Wait, the problem says: "Subsequent terms are produced by alternately adding and multiplying by each successive integer beginning with 1."

So starting with the first term, which is 0. Then the second term is produced by adding 1 to the first term. Third term is produced by multiplying the second term by 1. Fourth term is produced by adding 2 to the third term. Fifth term is produced by multiplying the fourth term by 2. Etc. So the operations are applied to the immediately preceding term. So each operation is applied to the previous term. Therefore, the steps are correct.

Another way to check: Let's list all the operations and terms with their indices:

n: term_n

1: 0

2: term1 +1 =0+1=1

3: term2 *1=1*1=1

4: term3 +2=1+2=3

5: term4 *2=3*2=6

6: term5 +3=6+3=9

7: term6 *3=9*3=27

8: term7 +4=27+4=31

9: term8 *4=31*4=124

10: term9 +5=124+5=129

Yes, that's correct. So term 10 is 129. Therefore, the answer is 129.

I think that's thorough. I can't see any mistakes in the calculations. So I'm confident the answer is 129.
Thus, the first term in the sequence that is greater than 125 is $\boxed{129}$.
</think>